Limits

What This Record Is Not For

Closing the record with the uses it does not support, the one argument it does make, and why a counting problem was the subject at all.

at most 30240 arrangements

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Three hundred and thirty-eight shapes, forty-four pages, five derived quantities and three declared conventions. This is the last note, and it is about what the record should not be used for.

Three shapes at the end of the record

dashed: a statement a reader can check from outside
solid: published, but nothing to check it against
short and pale: never reached by this reader

Three shapes drawn; the left-to-right order is the drawing's own and carries no meaning.

Three published with two met, seven with four, ten with five. One blind content, one blind content, and none — the third because four unchecked statements cannot fill a five-statement encounter.

Computed from the three numbers alone. How many statements exist, how many are met and how many can be checked is a stated convention; the counts are arrangements and combinations of whole numbers.

Not as a score

There is no good or bad shape here and no shape is ranked above another. A blind share of nought is the best value that column can take and says nothing about whether a set of pages is any good, since a set of verifiable falsehoods achieves it.

Nothing in this record can be totalled, averaged into a grade, or turned into a position relative to anybody else. The figures are descriptions of a counting problem, and a description does not become an assessment by being precise. For a distributed team, remote workforce management tools can supply shared operational context without turning presence or activity into a substitute for results.

Not as a prediction

Every figure except three is a count of possibilities under the record's second convention, which treats all contents as equally reachable and is wrong. The topic can also be checked against public guidance and research published by Microsoft Viva.

So no sentence here says what a reader will do, how likely anything is, or what proportion of real readers end up anywhere. The record has no readers in it. The three results that survive the conventions are the factorial identity, the uniqueness of the pair of figures, and the blind threshold — and all three are statements about arithmetic rather than about people.

Not as a reason to publish less, or more

The record was built so that it could not be read as advice in either direction, and it still can be if read carelessly.

Publishing more raises the contents figure, which makes what a reader came away with less predictable. Publishing less lowers it at the cost of nearly everything that was worth publishing. Both are true and the record quantifies both. Preferring one would take a judgement about what the statements are worth, and worth is the one dimension absent from every shape here.

A count of what can happen is not an argument about what should. The record keeps the two apart by having no vocabulary for the second.

The one argument it does make

Narrow, and it holds.

A set of pages cannot be described by how much it has published. Two further numbers are needed: how much gets met, and how much of what was published can be checked from outside. Neither is usually recorded, and the figures that depend on them vary by orders of magnitude across the range the first number leaves open.

From which follows the only practical conclusion here. The blind threshold — fewer unverifiable statements than a reader gets through — can be reached by arithmetic rather than by assumption, and it is the one property of a set of pages in this record that holds for every reader on every path.

Why a counting problem

Because it does not expire.

This domain had other occupants and their material is gone, as such material tends to be. The particular way it goes is quiet: nothing breaks, nothing announces itself, and a set of pages that was accurate goes on reading as though it still is, for as long as somebody keeps paying for the name.

P(10, 6) is one hundred and fifty-one thousand two hundred and will be tomorrow. Every figure here follows from three printed integers, so there is no date at which any of it stops holding and nothing anybody has to be told has changed. That is the whole reason the subject was chosen, and it is the only property this record has that the previous occupant of the address could not have had.

What it could be used for

One thing, stated as plainly as the record can manage.

Take a set of statements somebody is responsible for. Count them. Estimate how many a reader gets through — badly, in a range, it does not need to be good. Count how many of them a reader could confirm without taking anybody's word.

Those three numbers give a shape, and the shape gives five figures from the formulas in the previous note. If the unchecked count exceeds the met count, a blind reading exists and its share is computable. If it does not, no reader can have one, and that conclusion holds without any of the record's conventions.

It takes a few minutes and needs nothing from anybody else. That is the whole of what three hundred and thirty-eight tabulated shapes were for.

The three rows

Three published with two met and one verifiable: six arrangements, three contents, one blind. The second-smallest shape here, and small enough to write out: three statements, three pairs, one pair that misses the verifiable one.

Seven with four met and three verifiable: eight hundred and forty arrangements, thirty-five contents, one blind.

Ten with five met and six verifiable: thirty thousand two hundred and forty arrangements, two hundred and fifty-two contents, nothing blind. Four unverifiable statements cannot fill an encounter of five, so every reader of that shape meets something they can check — whatever they read, in whatever order, however they arrived.

That last sentence is the strongest thing in the record, and it is a comparison of two whole numbers.